Question 18
Let $ABC$ be an acute-angled triangle. Let $I_B$ and $I_C$ be the excentres of triangle $ABC$ opposite $B$ and $C$, respectively. Let $E$ be a point on line $BA$ such that $\angle EI_B B = 90^\circ$. Let $F$ be a point on line $CA$ such that $\angle CI_C F = 90^\circ$. Let the circle with centre $I_B$ and radius $I_B E$ intersect the segment $I_B B$ at $X$. Let the circle with centre $I_C$ and radius $I_C F$ intersect the segment $I_C C$ at $Y$.
Prove that BY is perpendicular to CX.
The excentre of a triangle ABC opposite vertex B is the centre of the circle that is tangent to line segment AC, to ray BA beyond A, and to ray BC beyond C. The excentre opposite C is similarly defined.
Prove that BY is perpendicular to CX.
The excentre of a triangle ABC opposite vertex B is the centre of the circle that is tangent to line segment AC, to ray BA beyond A, and to ray BC beyond C. The excentre opposite C is similarly defined.